Lectures on Numerical Radius Inequalities

Description

This book is a self-contained advanced monograph on inequalities involving the numerical radius of bounded linear operators acting on complex Hilbert spaces. The study of numerical range and numerical radius has a long and distinguished history starting from the Rayleigh quotients used in the 19th century to nowadays applications in quantum information theory and quantum computing. This monograph is intended for use by both researchers and graduate students of mathematics, physics, and engineering who have a basic background in functional analysis and operator theory. The book provides several challenging problems and detailed arguments for the majority of the results. Each chapter ends with some notes about historical views or further extensions of the topics. It contains a bibliography of about 180 items, so it can be used as a reference book including many classical and modern numerical radius inequalities. Preface Contents 1 Preliminaries 1.1 General Notation and Terminology 1.2 Elementary Functional Analysis 1.3 Functional Calculus for Selfadjoint Operators 1.4 Spectral Representation of Selfadjoint Operators 1.5 Block Matrices of Operators 2 Fundamental Numerical Radius Inequalities 2.1 Basic Properties of Numerical Range 2.2 Inequalities Concerning One Operator 2.3 Inequalities Related to Davis–Wielandt Radius 2.4 Inequalities Involving More Than One Operator 2.5 More Applications of Cartesian Decomposition 3 Bounds of the Numerical Radius Using Buzano's Inequality 3.1 Upper Bounds of the Numerical Radius 3.2 Generalized Buzano's Inequality 3.3 Vector Inequalities for Projections 4 p-Numerical Radius Inequalities of an n-Tuple of Operators 4.1 Basic Properties of p-Numerical Radius 4.2 Inequalities of wp for Pairs of Operators 4.3 Inequalities of wp for n-Tuples of Operators 4.4 Hypo-p-Numerical Radius 4.5 Reverse Inequalities for Hypo-Euclidean Numerical Radius 5 Numerical Radius Inequalities of Product of Operators 5.1 Inequalities Involving Adjoint Operators 5.2 Inequalities for Product of Two or More Operators 6 Numerical Radius of Operator Matrices and Applications 6.1 Basic Inequalities of 2times2 Operator Matrices 6.2 Inequalities of 2 times2 Off-Diagonal Operator Matrices 6.3 Bounds for the Zeros of a Polynomial 6.4 Bounded Region Containing All the Zeros of a Polynomial 7 Operator Space Numerical Radius of 2times2 Block Matrices 7.1 Complete Numerical Radius Norm 7.2 Inequalities for Operator Space Numerical Radius 8 A-Numerical Radius Inequalities in Semi-Hilbertian Spaces 8.1 Basic Definitions and Notations 8.2 Fundamental Properties 8.3 Inequalities Concerning One Operator 8.4 Inequalities Concerning Two Operators 8.5 Inequalities Concerning Block Matrices 9 Research Problems Appendix Bibliography